ASYMPTOTIC LIMITS AND STABILIZATION FOR THE 1D NONLINEAR MINDLIN-TIMOSHENKO SYSTEM

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1 J Syst Sci Comple 1) 3: 1 17 ASYMPTOTIC LIMITS AND STABILIZATION FOR THE 1D NONLINEAR MINDLIN-TIMOSHENKO SYSTEM F. D. ARARUNA P. BRAZ E SILVA E. ZUAZUA DOI: Received: 1 December 9 / Revised: 3 February 1 c The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 1 Abstract This paper shows how the so called von Kármán model can be obtained as a singular limit of a modified Mindlin-Timosheno system when the modulus of elasticity in shear tends to infinity, provided a regularizing term through a fourth order dispersive operator is added. Introducing damping mechanisms, the authors also show that the energy of solutions for this modified Mindlin-Timosheno system decays eponentially, uniformly with respect to the parameter. As, the authors obtain the damped von Kármán model with associated energy eponentially decaying to zero as well. Key words Mindlin-Timosheno system, singular limit, uniform stabilization, vibrating beams, von Kármán system. 1 Introduction The Mindlin-Timosheno system of equations is a widely used and fairly complete mathematical model for describing the transverse vibrations of beams. It is a more accurate model than the Euler-Bernoulli one, since it also taes into account transverse shear effects. The Mindlin-Timosheno system is used, for eample, to model aircraft wings see, e.g., [1]). For a beam of length L >, this one-dimensional nonlinear system reads as ρh 3 1 φ tt φ + φ + ) = in Q, ρh tt φ + ) [ η + 1 )] = in Q, 1) ρhη tt η + 1 ) = in Q, F. D. ARARUNA Departamento de Matemática Universidade Federal da Paraíba, , João Pessoa, PB, Brasil. fagner@mat.ufpb.br; pablo@dmat.ufpe.br. P. BRAZ E SILVA Departamento de Matemática, Universidade Federal de Pernambuco, , Recife, PE, Brasil. pablo@dmat.ufpe.br. E. ZUAZUA Ierbasque Research Professor, Basque Center for Applied Mathematics BCAM), Bizaia Technology Par, Building 5, E-4816, Derio, Basque Country, Spain. zuazua@bcamath.org. This paper is partially supported by INCTMat, FAPESQ-PB, CNPq Brazil) under Grant Nos. 3815/8- and 618/8-8, the MICINN Spain) under Grant No. MTM8-3541, the Advanced Grant FP NUMERIWAVES of the ERC, and the Project PI1-4 of the Basque Government, also dedicated to Prof. D. L. Russell on his 7th birthday.

2 F. D. ARARUNA P. BRAZ E SILVA E. ZUAZUA where Q =, L), T), the interval, L) is the segment occupied by the beam, and T is a given positive time. In system 1), subscripts mean partial derivatives. The unnowns φ = φ, t), =, t), and η = η, t) represent, respectively, the angle of rotation, the vertical displacement, and the longitudinal displacement at time t of the cross section located units from the end-point =. The constant h > represents the thicness of the beam which, in this model, is considered to be small and uniform with respect to. The constant ρ is the mass density per unit volume of the beam and the parameter is the so called modulus of elasticity in shear. It is given by the epression = Eh/ 1 + µ), where is a shear correction coefficient, E is the Young s modulus and µ is the Poisson s ratio, < µ < 1/. For more details concerning the Mindlin-Timosheno hypotheses and governing equations [ 3]. There is a large literature on this model, addressing problems of eistence, uniqueness and asymptotic behavior in time when some damping effects are considered, as well as some other important properties see [ 4] and references therein). When one assumes the linear filament of the beam to remain orthogonal to the deformed middle surface, the transverse shear effects are neglected, and one obtains the so called von Kármán system see [3]): ρh tt ρh3 1 tt + [ η + 1 )] = in Q, ρhη tt η + 1 ) = in Q. Note that neglecting the shear effects of the beam is formally equivalent to considering the modulus tending to infinity in system 1), since is inversely proportional to the shear angle. There is also an etensive literature about system ) see [ 3, 5 6] and references therein). In fact, since the pioneering wor of D. L. Russell and coworers summarized in his celebrated 1978 paper in SIAM Review [7], the control and stabilization of wave processes have undergone a significant progress. It would be impossible to summarize here the variety and the depth of the results developed after his influential wor. More recently, the subject has been developed even further using the so-called Hilbert Uniqueness Method HUM) developed by Lions and presented, in particular, in his SIAM Review article of 1988 [8]. Using HUM one can transform observability inequalities on the adjoint system that can be derived using multipliers, non-harmonic Fourier series techniques, microlocal analysis, Carleman inequalities, etc) into controllability ones by means of a fleible variational technique. This allows one to prove, for instance, the eistence of controls with minimal norm in different functional framewors. The same observability inequalities can then be used to build feedbac controllers and to get a variety of new stabilization results. There is also an etensive literature in this subject including wave, beam, plate, KdV, and Schrödinger equations, among other models considered. The present article is a further contribution in this contet, focusing mainly on the behavior of the eponential decay rate under a relevant singular perturbation in elasticity theory. More precisely, in the contet of systems of elasticity, it is well-nown that reduced models are often singular limits of more complete ones. The issue of the dependence of the decay rate or, more generally, the stabilization properties with respect to this singular parameter is also an interesting subject that has been object of intensive research. The present paper is devoted to addressing this issue in the contet of nonlinear models for vibrating beams. Our interest is to analyze the asymptotic limit of the nonlinear Mindlin-Timosheno system 1), when the shear modulus tends to infinity. This problem was mentioned in [3], where it was conjectured that the system 1) approaches, in some sense, the von Kármán system ) as. In order to achieve our results, we perturb system 1) by a term of the type /. More )

3 ASYMPTOTICS AND STABILIZATION FOR THE MINDLIN-TIMOSHENKO SYSTEM 3 precisely, instead of system 1) we consider the modified system: ρh 3 1 φ tt φ + φ + ) = in Q, ρh tt φ + ) [ η + 1 )] + 1 = in Q, ρhη tt η + 1 ) = in Q, and show the following: i) The system 3) converges to system ) as the parameter. ii) Adding appropriate damping terms, there is a uniform with respect to ) rate of decay for the total energy of the solutions of 3) as t. As a consequence of this analysis, we obtain a decay rate for the total energy of the solutions of the von Kármán system as t ) as a singular limit of the uniform with respect to ) decay rate of the energy of the Mindlin-Timosheno system. Note, however, that our results need the fourth order regularizing term in the component for compactness purposes. Reproducing the results of this paper without that regularizing term is an interesting open problem. These results are proved under suitable boundary and initial conditions that we shall mae precise below. The connections between these two systems and the asymptotic limit problem of passing to the limit as tends to infinity have been recently investigated in a number of different situations. For the linear case, that is, in the absence of the term [ )] η + 1, and without the equation for η, it was proved in [3] see also [9]) that the linear Mindlin-Timosheno system ρh 3 1 φ tt φ + φ + ) =, ρh tt φ + ) =, approaches, as, the Kirchhoff equation ρh tt ρh3 1 tt + =, 5) under various boundary conditions. Controllability properties of these linear systems have also been studied. More precisely, using Fourier series decompositions and filtering techniques, it was shown in [9] that the eact controllability of the Kirchhoff system may be obtained as the limit, when, of a partial controllability property for the Mindlin-Timosheno system. Note however that these techiques, based in the Fourier analysis, cannot be applied in the present nonlinear contet. In [4], the authors proved the eistence of a compact global attractor for system 4), for the two dimensional case, with nonlinear feedbac forces. They also showed the limit of this attractor, as, to be closely related to the attractor of the limit system 5). The problem of singular perturbations and stabilization related to the nonlinear von Kármán model has also been intensively investigated. We refer, for instance, to [1 13], where these issues are addressed under various boundary conditions. In the present paper we consider system 3) under Dirichlet boundary conditions for φ, clamped ends for, and Neumann boundary conditions on η: φ, ) = φl, ) = on, T),, ) = L, ) =, ) = L, ) = on, T), 6) η, ) = η L, ) = on, T), 3) 4)

4 4 F. D. ARARUNA P. BRAZ E SILVA E. ZUAZUA and the initial data { {φ, ),, ),η, )} = {φ,, η } in, L), {φ t, ), t, ),η t, )} = {φ 1, 1, η 1 } in, L). 7) In Section, we analyze the well-posedness of the system 3), 6), 7) in the corresponding finite energy space. In Section 3, we rigorously study the limit behavior of the Mindlin-Timosheno system as towards the von Kármán system. More precisely, we prove that solutions {φ,, η} of 3), 6), 7) converge to {,, η} as, where {, η} solves system ) with the same boundary conditions for and η. In Section 4, adding damping terms, we prove an uniform in ) eponential decay property for the solutions of 3), 6), 7). Finally, in Section 5, we briefly discuss some related issues and open problems. Global Well-Posedness In this section, for the sae of completeness, we study eistence and uniqueness of solutions for system 3), 6), 7). First of all, we introduce the Hilbert space X = H 1, L) L, L) H, L) L, L) V H, where V = H 1, L) H and H = {v L, L); v ) d = }, equipped with the norm {u 1, u, v 1, v, w 1, w } X = u 1 + ρh3 1 u + u 1 + v v 1 +ρh v + w 1 + ρh w, where denotes the norm in L, L). Theorem 1 If φ, φ 1,, 1, η, η 1 ) X, then problem 3), 6), 7) has a unique wea solution in the class {φ,, η} C [, );H 1, L) H, L) V ) C 1 [, ); [ L, L) ] ) H. 8) Moreover, the energy E t), given by E t) = 1 ρh 3 1 φ t t) + ρh t t) + ρh η t t) + φ t) + φt) + t) + η t) + 1 ) [ t)] + 1 t), 9) satisfies E t) = E ), t. 1) Remar 1 Note that the energy E t) defines a norm in X which is equivalent to X. Proof of Theorem 1 The problem 3), 6), 7) can be written as { Ut = AU + F U), U ) = U,

5 ASYMPTOTICS AND STABILIZATION FOR THE MINDLIN-TIMOSHENKO SYSTEM 5 where A = 1 ρh 3 1 ) 1 ρh 3 1 ρh 1 4 ρh, U = ρh F U) = [ η + 1 )] 1 ) ρh, and U = It is easy to see that the operator A : D A) X X with domain where D A) = [ H, L) H 1, L) ] H 1, L) [ H 4, L) H, L) ] H 1, L) W H 1, L), W = { v H, L); v ) = v L) = } is the infinitesimal generator of a semigroup of operators in X. Indeed, AU, U X = 1 d φ φ d φ φ d + φ + ) d + 1 φ + )φ d d On the other hand, given Φ = {α, β, γ, ζ, θ, λ} T X, the system φ φ 1 1 η η 1. η η d φ φ η η, φ + )φ + )d η η d =. A + I)U = Φ in, L) 11) has a unique solution U = {φ, φ,,, η, η } T D A). In fact, the system 11) reduces to φ φ = α, 1 ρh 3φ + 1 ρh 3 φ + 1 ρh 3 + φ = β, = γ, 1 ρh ρh ρh φ + = ζ, η η = θ, 1 ρh η + η = λ. 1)

6 6 F. D. ARARUNA P. BRAZ E SILVA E. ZUAZUA Solving system 1) is equivalent to finding such that φ ξ d + 1 {φ,, η} [ H, L) H 1, L) ] [ H 4, L) H, L) ] W σ d + φ + ) ξd + ρh3 1 φ + )σ d + ρh φξd = ρh3 1 σd = ρh α + β) ξd, ξ H 1, L), γ + ζ)σd, σ H, L), 13) and { η + ρhη = ρh θ + λ) in, L), η ) = η L) =. Define the bilinear form b : [H 1, L) H, L)] R by 14) b {φ, }, {ξ, σ}) = φ ξ d + ρh3 1 + and the linear one l : H 1, L) H, L) R by l {ξ, σ}) = ρh3 1 φξd + 1 φ + )ξ + σ )d + ρh α + β) ξd + ρh γ + ζ)σd. σ d σd, It is easy to see that b is continuous and coercive, and that l is continuous. So, due to the La-Milgram s theorem, there eists a unique {φ, } H 1, L) H, L) such that b {φ, }, {ξ, σ}) = l {ξ, σ}), {ξ, σ} H 1, L) H, L). Considering vectors of the form {ξ, }, with ξ H 1, L), and {, σ}, with σ H, L), one concludes that the pair {φ, } found above satisfies 13). Furthermore, from 1) 1 and 1) 3, one has {φ, } H 1, L) H, L). Applying classical elliptic regularity, it follows from 13) that {φ, } H, L) H 4, L). Classical elliptic theory applied for the operator d d +δi, δ >, with Neumann conditions, assures that 14) possesses only one solution in the class η W. In this way, one obtains the eistence of a unique solution {φ, φ,,, η, η } D A) for system 11). To prove eistence and uniqueness of local solutions for the nonlinear problem, it remains to show that F U) is locally Lipschitz continuous in X. In fact, if U = {φ, φ,,, η, η } T, Ũ = { φ, φ,,, η, η } T belong to X, we have F U) FŨ) = ρh f + g ), 15) X where f = [ η + 1 ) η + 1 )] and g = 1 ). Adding and subtracting the term η + 1 ) inside the norm f, one gets f L,L) η L η η,l) + 1 L + L. 16),L),L)

7 ASYMPTOTICS AND STABILIZATION FOR THE MINDLIN-TIMOSHENKO SYSTEM 7 Using the embedding of H 1, L) into L, L), one has from 16) that ) U f C U X, Ũ Ũ. 17) X X Using once again the embedding of H 1, L) into L, L), one also sees that ) U g C U X, Ũ Ũ. 18) X X Combining 15), 17) and 18), it follows that F U) is locally Lipschitz continuous in X. Global eistence in our case is a consequence of the total energy to be conservative, that is, E t) = or E t) = E ) for all t. This concludes the result. Remar Theorem 1 guarantees the eistence and uniqueness of finite-energy solutions for initial data {φ, φ 1,, 1, η, η 1 } X. In particular, we have considered η )d = η 1 ) d =. 19) This condition is not necessary to obtain a unique finite-energy solution. Indeed, integrating equation 3) 3 with respect to, we have which implies d dt η, t)d = η, t)d =, η ) d + t η 1 ) d. ) According to ), the unique solution of the system is easily constructed without the condition 19). We set φ = φ, =, η = 1 η ) d + t 1) η 1 ) d + η, L L where { φ,, η} is the unique solution provided by Theorem 1 associated to initial data {φ, φ 1,, 1, η, η 1 }, with η = η 1 L η ) d and η 1 = η 1 1 L η 1 ) d, which clearly satisfy 19). Even though the zero average condition 19) is not necessary to show eistence of solutions, it seems necessary to assure the eistence of a solution defined in the whole interval [, ) see 1)). This is why we require 19), since we are also interested in studying the asymptotic behavior of solutions as t see Section 4). 3 Asymptotic Limit In this section, we study the asymptotic limit of the solutions for the nonlinear Mindlin- Timosheno system 3), 6), 7) as.

8 8 F. D. ARARUNA P. BRAZ E SILVA E. ZUAZUA { Theorem Let φ,, η } be the unique solution of 3), 6), 7) with data {φ, φ 1,, 1, η, η 1 } X satisfying Then, letting, one gets φ + = in, L). ) { φ,, η } {,, η} wea in L, T; [ H 1, L)] V ), where {, η} solves the von Kármán system: ρh tt ρh3 1 tt + [ η + 1 )] = in Q, ρhη tt η + 1 ) = in Q,, ) = L, ) =, ) = L, ) = on, T), η, ) = η L, ) = ) on, T),, ) =, h 1 ) = 1 + h t 1 φ 1 in, L), η, ) = η, η t, ) = η 1 in, L). 3) Remar 3 Eistence and uniqueness of wea solutions for the limit system 3) was proved in [1]. More precisely, when {, 1, η, η 1 } H, L) L, L) V H, there eists a unique solution {, η} in the class {, t, η, η t } C [, );H, L) H1, L) V H), satisfying the variational formulation of system 3) ρh d dt t t), w) + ρh3 d 1 dt t),w ) + ρh d dt η t t),y) + t),w ) + t) η t) + 1 ) ) [ t)], w + η t) + 1 ) [ t)], y =, 4) for all {w, y} H, L) H1, L) and the initial conditions 3) 5, 3) 6. In 4),, ) represents the inner product in L, L). Furthermore, the energy Et) of system 3) Et) = 1 ρh t t) + ρh η t t) + ρh3 1 t t) + t) + η t) + 1 ) [ t)] 5) is conservative, that is, Et) = E), for all t [, T]. Remar 4 Note that, in order to fully identify the initial data of the solutions of the limit system 3) and, more precisely, to determine the initial data of t, an elliptic equation has to be solved. Namely, the initial datum for the velocity t in 3) 5 is determined by solving the elliptic equation t, ) H 1, L); ρh t ) ρh3 1 t ) = ρh 1 + ρh3 1 φ 1,

9 ASYMPTOTICS AND STABILIZATION FOR THE MINDLIN-TIMOSHENKO SYSTEM 9 as the proof of the theorem will show. More precisely, this elliptic equation can be written in the variational form ρh3 1 t), w ) ρh t ), w) = ρh3 1 φ 1, w ) ρh 1, w), w H 1, L), where the term φ 1, which is an element of H 1, L), is not the derivative of φ 1 in the sense of transposition, but rather the linear mapping which, when acting on any element w of H 1, L), yields the value φ 1, w ). The same can be said about t ) which represents the element of H 1, L) yielding t ), w ). Remar 5 According to Remar, one sees that condition 19) is not necessary to obtain the von Kármán system 3) as an asymptotic limit as ) of the perturbed Mindlin- Timosheno system 3), 6), 7). Here, we assume again the average zero condition 19) only to assure the eistence of solutions defined in the whole interval [, ) to be able to study its asymptotic behavior. This study will be carried out in the net section. Proof of Theorem For each > fied, let {φ,, η } be the solution of system 3), 6), 7) with data {φ, φ 1,, 1, η, η 1 } X. We are interested in the asymptotic limit, as, of system 3), 6), 7). Since the initial data {φ, φ 1,, 1, η, η 1 } satisfy the condition ), one has, due to the conservation of energy 1), that E ) C, >. 6) Therefore, the sequences in ) φ ), 1 ), η ) 7) are bounded in L, T; H 1, L) ), L, T; H, L) ), and L, T; V ), respectively, and ) ) ) φ t, t, η t, φ + ) ), η + 1 ) ) 8) remain bounded in L, T; L, L) ). Etracting subsequences, without changing notation, one gets 1 α wea in L, T; H, L)), 9) { φ,, η } {φ,, η} wea in L, T; [ H 1, L) ] V ), 3) with Moreover, φ + =. 31) { φ t, t, ηt } {φt, t, η t } wea in L, T; [ L, L) ] ) 3, 3) and η + 1 ) ξ wea in L, T; L, L) ). 33) Now, using 9), we also get 1 = 1 1 wea in L, T; H, L)). 34)

10 1 F. D. ARARUNA P. BRAZ E SILVA E. ZUAZUA According to 6), we have, for 1, the following estimate: 1 φ t) + t) H 1 φ,l) t) + 1 t) C. 35) The character C stands for a generic positive constant which may vary from line to line unless otherwise stated). By interpolation, we can deduce that φ t) + t) H 1/,L) C 1/4 φ t) + t) 1/) ) 1/4 φ t) + t) 1/ H 1,L). 36) Combining 6), 35), and 36), we obtain t) H C. 37) 1/,L) We also have by 6) that t) H 1/,L) C H1,L) C φ + + φ ) C. 38) Since the norms u H 3/,L) and ) 1/ u H 1/,L) + u H 1/,L) are equivalent see [14, Theorem 9.7]), we get by 37) and 38) that the sequence ) in L, T; H 3/, L)) W 1,, T; L, L) ) is uniformly bounded with respect to. In this way, due to the Aubin-Lions compactness theorem see [15, Corollary 4]), we obtain ) strongly in L, T; H 3 δ, L), 39) for any δ > which, by Sobolev s embeddings, implies the strong convergence in L, T; L p, L)), for all 1 p <. Combining 3) and 39), it follows that ξ = η + /, and [ η + 1 ] ) [η + 1 ] wealy in L Q). 4) For {z, w, y} H 1, L) H, L) H1, L) satisfying the variational formulation of 3), 6), 7) is simply z + w =, 41) ρh 3 d φ 1 dt t t),z ) + ρh d dt t t),w ) + ρh d η dt t t),y ) + ) φ t),z + t) η t) + 1 [ t) ] ) ), w + η t) + 1 [ t) ] ), y + 1 t),w ) =. 4) Using convergences 3), 3) 34) and 4) in equation 4), and applying identities 31) and 41), one obtains the wea formulation of the system 3) given in 4). To finish the proof,

11 ASYMPTOTICS AND STABILIZATION FOR THE MINDLIN-TIMOSHENKO SYSTEM 11 it remains to identify the initial data of the limit system. In view of the convergences 3), 3), and classical compactness arguments, one has {, η } {, η} in C [, T]; [L, L)] ). Then, {, ),η, ) } {, ),η, )} in [L, L)], which combined with 7) 1, guarantees that {, ),η, )} = {, η }. In order to identify { t, ),η t, )}, multiply both sides of equation 4) by the function θ δ H 1, T) defined by { t + 1, if t [, δ], θ δ t) = δ, if t δ, T], and integrate by parts to obtain ρh3 1 φ 1, z) + ρh3 1δ ρh η 1, y) + ρh φ t t),z ) dt ρh 1, w) + ρh δ t t),w ) dt η t t), y ) ) dt + φ t), z θδ t)dt + 1 δ ) t),w θδ t)dt + t) η t) + 1 [ t) ] ) ), w θ δ t)dt + η t) + 1 [ t) ] ), y θ δ t) dt =. 43) Passing to the limit as in the last equation, and using 3), 3) 34) and 4), one obtains ρh 3 1 φ 1, w ) ρh 1, w) ρh η 1, y) + ρh3 1δ + ρh δ + t t),w) dt + ρh t) η t t),y)dt + η t) + 1 [ t)] ), w ) θ δ t)dt + t t),w ) dt t),w )θ δ t)dt η t) + 1 [ y)], y ) θ δ t)dt =. On the other hand, multiplying Equation 4) by θ δ and integrating in time we get the identity ρh3 1 t ),w) + ρh t ),w) + ρh η t ),y) = ρh3 1 φ 1, w) + ρh 1, w) + ρh η 1, y), {w, y} H, L) H1, L). 44) In this way, we conclude that h 1 ) t ) = 1 + h 1 φ 1, and η t ) = η 1. 4 Uniform Stabilization The aim of this section is to obtain the eponential decay for the energy 5) associated to the solution of the von Kármán system ρh tt ρh3 1 tt + [ η + 1 )] + t t = in Q, ρhη tt η + 1 ) + η t = in Q, 45), ) = L, ) =, ) = L, ) = on, T), η, ) = η L, ) = on, T), {, ), t, ),η, ),η t, )} = {, 1, η, η 1 } in, L),

12 1 F. D. ARARUNA P. BRAZ E SILVA E. ZUAZUA as a limit as ) of the uniform stabilization of the perturbed Mindlin-Timosheno system ρh 3 1 φ tt φ + φ + ) + φ t = in Q, ρh tt φ + ) [ η + 1 )] t = in Q, ρhη tt η + 1 ) + η t = in Q, 46) φ, ) = φl, ) = on, T),, ) = L, ) = on, T), η, ) = η L, ) = on, T), {φ, ),, ),η, )} = {φ,, η } in, L), {φ t, ), t, ),η t, )} = {φ 1, 1, η 1 } in, L). Analogously to the proof of Theorem, the system 45) can be obtained as a limit, when, of system 46). Notice that its energy given by 9) obeys the energy dissipation law E t) = φ t t) + t t) + η t t) ), 47) that is, E t) is a non increasing function. We will show that this energy decays eponentially as t ) uniformly with respect to. More precisely, the following result holds. Theorem 3 Let {φ,, η} be the global solution of system 46) for data {φ, φ 1,, 1, η, η 1 } in X satisfying ). Then there eists a constant ω > such that E t) 4E )e ω t, t. 48) Remar 6 As a consequence of inequality 48), letting one recovers the eponential decay of the energy E t), associated to system 45), and given by 5). This is in agreement with the results obtained in [1]. Remar 7 Note that the zero average condition 19) is necessary to prove Theorem 3, since one needs to assure all time eistence of solutions. It is also important to note that the zero average property is preserved for the damped system 46), that is, its solutions have zero average as well. Indeed, integrating equation 46) 3 with respect to, we get which implies η, t)d = d dt η, t) d + d dt η )d + η, t)d =, ) L η 1 )d η 1 ) d e t =. Proof of Theorem 3 For each 1 fied, let {φ,, η } be the solution of system 3), 6), 7) with data {φ, φ 1,, 1, η, η 1 } X. From now on, in this proof, we will omit the inde of the solution to simplify the notation. For an arbitrary λ >, define the perturbed energy G λ t) = E t) + λf t), 49) where F is the functional ) ρh 3 F t) = θ 1 φ t t), φt) + θ ρh t t), t)) + θ ρhη t t),η t)), 5) for θ > a constant to be chosen later on. Let us bound each term on the right hand side of identity 5) by an epression involving the energy 9).

13 ASYMPTOTICS AND STABILIZATION FOR THE MINDLIN-TIMOSHENKO SYSTEM 13 ) Analysis of θ ρh 3 1 φ t t),φt). Using Poincaré inequality, one gets ) ρh 3 θ 1 φ t t),φt) θ ρh 3 ) L ρh φ t t) + φ t) ρh 3 L θ 1 E t). 51) Analysis of θ ρh t t), t)). Using Poincaré inequality as well, one gets θ ρh t t), t)) θρh L t t) φt) + t) + ) L φ t) ρhl ) θ + ρhl E t). 5) Analysis of θ ρhη t t),η t)). One has θ ρhη t t),η t)) θρh η t t) η t). 53) Notice that the initial data satisfy ). Thus, one can guarantee, due to 47), that the sequence φ ) is bounded in L, T; H 1, L)), and that φ + ), η + 1 ) ) are bounded in L, T; L, L) ). Consequently, the sequence ) is bounded in L, T; L, L) ). So, the sequence ) ) remains bounded in L, T; L 1, L) ). One also has that η ) is bounded in L, T; L 1, L) ). Since η t) V Remar 7), one uses the Poincaré-Wirtinger inequality see [16]) and gets η t) L η t) L 1,L) L η t) + 1 [ t)] + On the other hand, L [ t)] L 1,L) L t) t) L L [ t)] L 1,L). 54) φt) + t) + ) L φ t) φt) + t) + ) L φ t) 1 + L) L E t) φt) + t) + ) L φ t) 1 + L) LC φt) + t) + ) L φ t), 55) where C is given in 6). Therefore, it follows by 53) 55) that θ ρhη t t),η t)) θ 1 + L) L C ρh η t t) φ t) + θlρh η t t) η t) + 1 [ t)] +θ 1 + L) LC ρh η t t) φt) + t) θ [1 + L) ρhl C + ρhl L) ] ρhlc E t). 56)

14 14 F. D. ARARUNA P. BRAZ E SILVA E. ZUAZUA According to the bounds 51), 5), and 56), one has F t) C 1 E t), 57) where C 1 = θ ρhl[1+3 L+ 1/1h+1+L)1+ L) C ]. Now, using 49) and 57), one has G λ t) E t) λ F t) λc 1 E t), which is equivalent to Taing < λ 1/C 1, one gets 1 λc 1 )E t) G λ t) 1 + λc 1 )E t). E t) G λ t) E t). 58) Differentiating the functional F and using the equations in 46), one obtains F t) = θ φ t) θ φt) + t) θ η + 1 [ t)] + θρh3 1 φ t t) + θρh t t) + θρh η t t) θ φ t t),φt)) + θ t t), t)) θ η t t), η t)). 59) We bound each term that appears on the right hand side of the identity 59) separately. Analysis of θ φ t t),φt)). θ φ t t),φt)) C φ t t) + ξ φ t) C φ t t) + ξe t), 6) where C = θ L/ξ and ξ > is a real number to be appropriately chosen. Analysis of θ t t), t)). θ t t), t)) C 3 t t) + ξ φ t) + ξ φt) + t) where C 3 = θ L 1 + L)/ξ. C 3 t t) + ξe t), 61) Analysis of θ η t t),η t)). It follows by inequalities 54) and 55) that θ η t t),η t)) θ η t t) η t) θ 1 + L) L C η t t) φ t) + θl η t t) η t) + 1 [ t)] +θ 1 + L) LC η t t) φt) + t) C 4 η t t) + ξe t), 6) where C 4 = θ L[L L) 3 C ]/ξ.

15 ASYMPTOTICS AND STABILIZATION FOR THE MINDLIN-TIMOSHENKO SYSTEM 15 Using bounds 6), 61), and 6), one obtains, from 59), F t) θ φ t) θ φt) + t) θ η + 1 ) ) θρh C φ t t) Therefore, + θρh + C 3 ) t t) + θρh + C 4 ) η t t) + 3ξE t). F t) θ 3ξ)E t) + C 5 φ t t) + C 6 t t) + C 7 η t t), 63) where C 5 = θρh 3 /1 ) + C, C 6 = θρh + C 3, and C 7 = θρh + C 4. Considering the derivative of the epression 49), and observing 47) and 63), one has G λ t) λθ 3ξ)E t) 1 λc 5 ) φ t t) 1 λc 6 ) t t) 1 λc 7 ) η t t). Taing λ = min {1/C 1, 1/C 5, 1/C 6, 1/C 7 } and ξ < θ/3, one obtains, according to 58), where ω = λθ 3ξ). In this way, G λ t) ω G λ t), t, G λ t) G λ )e ω t, t. 64) Combining 58) and 64), one gets 48). This finishes the proof. For the original dissipative Mindlin-Timosheno system ρh 3 1 φ tt φ + φ + ) + φ t = in Q, ρh tt φ + ) [ η + 1 )] + t = in Q, ρhη tt η + 1 ) + η t = in Q, φ, ) = φl, ) = on, T), 65), ) = L, ) = on, T), η, ) = η L, ) = on, T), {φ, ),, ),η, )} = {φ,, η } in, L), {φ t, ), t, ),η t, )} = {φ 1, 1, η 1 } in, L), one can proceed analogously to the proof of Theorem 3 to get the following Theorem 4 Let {φ,, η} be the global solution of system 65) for data {φ, φ 1,, 1, η, η 1 } in X satisfying ). Then there eists a constant ω > such that the energy E t) = 1 ρh 3 1 φ t t) + ρh t t) + ρh η t t) + φ t) + φt) + t) + η t) + 1 ) [ t)]

16 16 F. D. ARARUNA P. BRAZ E SILVA E. ZUAZUA satisfies E t) 4E )e ω t, t. Remar 8 Even though Theorem 4 provides a uniform in ) stabilization result, passing to the limit in this system as is an open problem, because of the lac of compactness of the quadratic term on. 5 Further Comments and Open Problems In this section we briefly discuss some possible etensions of our results and also indicate open problems on the subject. 1) The methods of this paper wor in the contet of model 3) in which a fourth order dispersive term has been added to ensure the compactness of the families of solutions as the singular parameter tends to infinity. It would be interesting to analyze whether the same results hold without that regularizing term. The difficulty, of course, consists in getting enough compactness to pass to the limit in the nonlinear term. ) It would be interesting to analyze whether the results of this paper hold for different boundary conditions. 3) In the proof of Theorems and 3, we have considered the case when the initial data φ, φ 1,, 1, η and η 1 are fied. The same results hold if we consider the case where they do depend on, provided we assume that φ, φ 1,, 1, η and η 1 are such that the initial energy E ) remains bounded and {φ, φ 1,, 1, η, η 1 } converges wealy to {φ, φ 1,, 1, η, η 1 } in the corresponding spaces. 4) One could address the problem of uniform stabilization with damping terms that are only active on the rotation angle and longitudinal displacement equations in the Mindlin-Timosheno system, that is, systems 46) and 65) without adding the dissipative term t. This is a natural choice, since, in the limit, the dissipative term φ t generates the damping t which acts in the vertical displacement equation of the von Kármán system and suffices to guarantee the eponential decay of the limit system. One could epect the energy of the approimating system to decay uniformly, with respect to the parameter, as t. References [1] J. F. Doyle, Wave Propagation in Structures, Springer-Verlag, New Yor, [] J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM, [3] J. E. Lagnese and J. L. Lions, Modelling Analysis and Control of Thin Plates, RMA 6, Masson, Paris, [4] I. Chueshov and I. Lasieca, Global attractors for Mindlin-Timosheno plates and for their Kirchhoff limits, Milan J. Math., 6, 74: [5] A. Favini, M. A. Horn, I. Lasieca, and D. Tartaru, Global eistence, uniqueness and regularity of solutions to a von Kármán system with nonlinear boundary dissipation, Diff. Integ. Eqns, 1996, 9: [6] J. E. Lagnese and G. Leugering, Uniform stabilization of a nonlinear beam by nonlinear boundary feedbac, J. Diff. Eqns, 1991, 91: [7] D. L. Russell, Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions, SIAM Rev., 1978, : [8] J. L. Lions, Eact controllability, stabilizability and perturbations for distributed systems, SIAM Rev., 1988, 3: 1 68.

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